/// <summary>
/// Preconditioned Conjugate Gradient Method <see cref="SolveConjugateGradient(Microsoft.Msagl.Core.Layout.ProximityOverlapRemoval.ConjugateGradient.SparseMatrix,Microsoft.Msagl.Core.Layout.ProximityOverlapRemoval.ConjugateGradient.Vector,Microsoft.Msagl.Core.Layout.ProximityOverlapRemoval.ConjugateGradient.Vector,int,double)"/>
/// Preconditioner: Jacobi Preconditioner.
/// This method should generally be preferred, due to its faster convergence.
/// </summary>
/// <param name="A"></param>
/// <param name="b"></param>
/// <param name="x"></param>
/// <param name="iMax"></param>
/// <param name="epsilon"></param>
/// <returns></returns>
public static Vector SolvePrecondConjugateGradient(SparseMatrix A, Vector b, Vector x, int iMax, double epsilon)
{
//create Jacobi (diagonal) preconditioner
// M=diagonal(A), M^-1=1/M(i,i), for i=0,...,n
// since M^-1 is only applied to vectors, we can just keep the diagonals as a vector
Vector Minv = A.DiagonalPreconditioner();
Vector r = b - (A * x);
Vector d = Minv.CompProduct(r);
double deltaNew = r * d;
double normRes = Math.Sqrt(r * r) / A.NumRow;
double normRes0 = normRes;
int i = 0;
while ((i++) < iMax && normRes > epsilon * normRes0)
{
Vector q = A * d;
double alpha = deltaNew / (d * q);
x.Add(alpha * d);
// correct residual to avoid error accumulation of floating point computation, other methods are possible.
// if (i == Math.Max(50, (int)Math.Sqrt(A.NumRow)))
// r = b - (A*x);
// else
r.Sub(alpha * q);
Vector s = Minv.CompProduct(r);
double deltaOld = deltaNew;
normRes = Math.Sqrt(r * r) / A.NumRow;
deltaNew = r * s;
double beta = deltaNew / deltaOld;
d = s + beta * d;
}
return(x);
}
